The world around us is full of objects that due to their physical properties are nonrigid and therefore can be deformed and bent. Nonrigid shapes appear at all scales in nature -- from our body, its organs and tissues, to tiny bacteria and microscopic protein molecules. Being so ubiquitous, such shapes are often encountered in pattern recognition and computer vision applications. The richness of the possible deformations of nonrigid shapes appears to be a nightmare for a pattern recognition researcher, who faces a vast number of degrees of freedom when trying to analyze them. In this talk, we address two major problems in the analysis of nonrigid shapes: similarity and correspondence. First, we present a construction of similarity criteria for deformation-invariant shape comparison based on intrinsic geometric properties of the shapes, and visualize them in a three-dimensional face recognition application. We consider the interesting particular case of shape self-similarity or intrinsic symmetry -- a nonrigid generalization of the well-known notion of Euclidean symmetry. Next, we extend the problem of similarity computation to shapes which have similar parts but are dissimilar when considered as a whole. Finally, we show that as a byproduct of the nonrigid similarity problem, correspondence between shapes is obtained. We use this correspondence to construct a calculus of nonrigid shapes, which is demonstrated in computer graphics application.